Such a device is known from EP 344 859. The known device serves to dispense heated milk and further contains a frothing device, by means of which the milk can additionally be frothed. Most different operating principles of frothing devices for milk are known. In the device described, for example, the milk is drawn in from a storage tank by means of the Venturi principle. This is accomplished through a supply line in the form of a hose, the outlet end of which is connected to the frothing device and the inlet end of which is simply inserted into a storage tank for milk. A problem involved by all milk handling devices is the cleaning. Milk is a perishable product, so that the components coming into contact with milk should be cleaned as often and as thoroughly as possible. Inside the heating and frothing device this is not so problematical because steam is used therein, so that the frothing device is relatively easy to clean with a pure steam blast, without the introduction of milk, whereby, at least until the next thorough cleaning procedure involving the disassembly of the components and/or the use of cleaning agents, the growth of bacteria is safely prevented. However, as there is no natural way for the steam into the supply line, the supply line can, in this manner, not or at least not sufficiently be cleaned. In connection with the known device it is, therefore, proposed to place the inlet end of the supply line into a cleaning liquid in the meantime and, by operating the device, to draw in the cleaning liquid instead of milk through the supply line. However, this is complicated and may not lead to the desired result. To thoroughly clean the supply line, also hot, nearly boiling water could be used, which is difficult to handle, however, and is not always available either. Cold or merely warm water does not bring about the desired cleaning result. If chemical cleaning agents are used, a complicated rinsing has to be performed afterwards to make sure that all residues are removed from the device.